Issue 43

P. Zampieri et alii, Frattura ed Integrità Strutturale, 43 (2018) 182-190; DOI: 10.3221/IGF-ESIS.43.14 183 Referring to the main studies about horizontal displacement available in literature, Ochsendorf [22] analyzed the formation of cracks of a masonry arch in relation to the horizontal thrust and the distinctive geometrical characteristics of the structure, that are thickness t, radius R and angle of embrace β, through thrust line analysis. Coccia et al. [23] reviewed the Ochsendorf’s work suggesting an innovative method in studying these phenomenons based on the implementation of kinematic theorem to deformed configuration of the arch. In particular, they showed that the considered collapse configuration for the masonry arch, then the position of the cracks, depend on the displacement imposed to the supports. Hence all the studies conducted focused on the collapse mechanism for the horizontal displacement, the aim of this paper is to analyze others settlements condition. In particular, an imposed displacement with two different direction components that are horizontal and vertical. For these reasons, the purpose of this work is to amplify the already avaiable investigations in settlement springing of masonry archs analyzing other settlement condition, to develop the arch behavior knowledge. Throughout limit analysis and experimental testing, the structural behaviour of masonry arches subjected to a d k settlement of springing on an inclined direction α of 45° (Fig. 1) is investigated. Moreover, knowing that the cracks position in a collapse configuration may vary increasing the settlement, as already stated throughout FEM and experimental analysis, this behaviour has been analyzed for a 45° direction settlement direction by means of a Limit Analysis then compared with the field test. The analysis procedure used to assess the thrust line and reaction forces of the springing will be described in detail for each incremental displacement value d k , until reaching the condition of complete collapse of the arch. This procedure uses: i) limit analysis in the hypotheses of significant arch displacements, and ii) analysis of thrust lines. Following this, the results of experimental testing on a masonry arch with mortar joints subjected to one of two springings incremental settlement (inclination equal to 45°) will be presented. As will be illustrated, the comparison of analytical results and experimental results demonstrates the great predictive ability of the calculation model developed. ANALYSIS OF MASONRY ARCHES WITH NON - HORIZONTAL SPRINGING SETTLEMENT hen a masonry arch loses a degree of freedom (and may be displaced along a given direction α), a collapse mechanism is created, with three cracking hinges (Fig. 1) and the thrust line within the form of the arch is tangential to the edge at the three hinge points. This three-hinge mechanism is created as a result of small movements along the α direction. Thus, assuming that the displacement in the α direction that triggers the collapse mechanism is null, it is possible to: remove the constraint condition in direction α and replace it with an equivalent reaction force R α, 0 (Fig.1); define a three-hinge collapse mechanism and an associated field of virtual displacements (Fig. 1) and apply PVW using Heyman’s hypotheses [1]. For an arch consisting of n blocks, assuming any initial position of hinges 1, 2, 3 (Fig. 1), it is possible to define the external work of gravitational forces g i and the force R α,0 as:   ,0 s 0 0 1 , 0 n e i i i L g v R x y           (1) where: g i is the gravitational force applied to each i-rigid block of the arch; v i is the vertical virtual displacement due to g i applied to each i-rigid block of the arch. s(X 0 , Y 0 ) is the virtual displacement of the settled springing. From (1), the value of the support reaction force R α0 can be obtained, as follows:   1 ,0 s 0 0 , n i i i g v R x y        (2) Once the value of R α0 is known, it is possible to: define the thrust line following the procedure shown in Zampieri et al 2016 [10], determine a new updated position of the three cracking hinges and apply PVW again. Repeating this process, convergence is reached when the solution sought is both statically and kinematically admissible at the same time. This W